Torsional angle energy function

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

The crystal and molecular structure data of the three Se8 forms listed in Table II have been determined by X-ray diffraction (27-29, 31). a-, /3-, and y-cyclooctaselenium crystallize in the same space group but differ in the packing of the molecules (see Fig. 4). The average bond distances, bond angles, and torsional angles of the Se8 molecules are identical within the limits of the standard deviation. The torsional angle of 101° is close to the value of 99° observed in the case of Sg (36) and obviously corresponds to the minimum of the torsional potential energy function. The shortest intermolecular distance has been observed in the case of -y-Se8 the value of 334.6 pm is even smaller than the shortest intermolecular contact in orthorhombic cy-clooctasulfur, S8 [337 pm (33)]. 

The torsional angles of gauche conformers of a number of molecules are included in this study. However, the prediction of torsional potential energy functions for these molecules are not included although geometrical structures at conformation minima are indeed related to torsional potentials. Further systematic study of these problems is required. 

The results for the C71 chains also show that the performance of off-lattice simulations must be assessed by looking at all possible aspects of the simulation. Often used criteria, such as internal energy variation, torsional angle distribution function, and radial distribution function [38, 39,45] do not by themselves provide reliable performance indicators. 

Generally, only a small number of scaling constants are needed. For example, ten scaling constants and reference values were derived for hydrocarbons from comparisons of gas phase structures, conformational energies, rotational barriers, and vibrational frequencies measured by experiment and calculated by the QMFF. For the bond and bond angle energy functions in equation (1) the same scale factor is multiplied by the QMFF quadratic, cubic, and quartic force constants. Similarly, the same scale factor is used for the one-, two-, and threefold torsion force constants, and a single scale factor is used for all cross terms. 

Figure 2-108. Derivation of a syrMbolic potential energy function from the torsion angle distribution of a torsion fragment.

Figure 2-108 shows the correspondence between a histogram and the derived empirical energy function for the torsion angle fragment C-N H)-C(H)(H -C. 

Sketch the form of the potential energy as a function of torsional angle cj) for the torsional vibration in (a) ethane, (b) CH3NO2, (c) 2-fluorophenol, (d) CH2FOH, and... 

Fig. 3.1. Potential energy as a function of torsion angle for ethane.

As mentioned in Section 2.2.3, the out-of-plane energy may also be described by an improper torsional angle. For the example shown in Figure 2.6, a torsional angle ABCD may be defined, even though there is no bond between C and D. The out-of-plane oop may then be described by an angle for example as a harmonic function... 

Table 2.3 Comparison of functional forms used in common force fields. The torsional energy, [ors is in all cases given as a Fourier series in the torsional angle...

In Figure 2 we have plotted the energy stored in the bond lengths,bond angles, and torsional angle as a function of chain extension for the three values of the constraining force shown in Figure 1. At a value of 7c =... 

Figure 2. The energy of the torsional angle, bond angles, and bond lengths are shown as a function of dj for the four atom chain with values of 7 equal to 5.0 x 10, 5.0 x 10 and 5.0 X 10 kg/mol-ns. (Reproduced with permission from ref. 13. Copyright 1988 Wiley.)...

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